Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 87.1% → 99.8%

Time: 13.2s

Alternatives: 15

Speedup: 1.4×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-5)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (+
        (* 0.016666666666666666 (* J (pow l 5.0)))
        (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * ((0.016666666666666666 * (J * pow(l, 5.0))) + ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * ((0.016666666666666666 * (J * Math.pow(l, 5.0))) + ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-5):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * ((0.016666666666666666 * (J * math.pow(l, 5.0))) + ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-5))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))) + Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-5)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * ((0.016666666666666666 * (J * (l ^ 5.0))) + ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-5]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.00000000000000016e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.00000000000000016e-5

   1. Initial program 71.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-5)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-5):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-5))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-5)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-5]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.00000000000000016e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.00000000000000016e-5

   1. Initial program 71.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(t_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-5)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       J
       (*
        (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))
        (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e-5)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)) * cos((K * 0.5))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e-5)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)) * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e-5):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (J * (((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)) * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e-5))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)) * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e-5)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (J * (((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)) * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.00000000000000016e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.00000000000000016e-5

   1. Initial program 71.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 99.8%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{if}\;t_0 \leq -0.96:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.85:\\ \;\;\;\;U\\ \mathbf{elif}\;t_0 \leq -0.35:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (+ U (* (* J (* l (pow K 2.0))) -0.25))))
   (if (<= t_0 -0.96)
     t_1
     (if (<= t_0 -0.85) U (if (<= t_0 -0.35) t_1 (+ U (* J (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + ((J * (l * pow(K, 2.0))) * -0.25);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = t_1;
	} else if (t_0 <= -0.85) {
		tmp = U;
	} else if (t_0 <= -0.35) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + ((j * (l * (k ** 2.0d0))) * (-0.25d0))
    if (t_0 <= (-0.96d0)) then
        tmp = t_1
    else if (t_0 <= (-0.85d0)) then
        tmp = u
    else if (t_0 <= (-0.35d0)) then
        tmp = t_1
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + ((J * (l * Math.pow(K, 2.0))) * -0.25);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = t_1;
	} else if (t_0 <= -0.85) {
		tmp = U;
	} else if (t_0 <= -0.35) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + ((J * (l * math.pow(K, 2.0))) * -0.25)
	tmp = 0
	if t_0 <= -0.96:
		tmp = t_1
	elif t_0 <= -0.85:
		tmp = U
	elif t_0 <= -0.35:
		tmp = t_1
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(Float64(J * Float64(l * (K ^ 2.0))) * -0.25))
	tmp = 0.0
	if (t_0 <= -0.96)
		tmp = t_1;
	elseif (t_0 <= -0.85)
		tmp = U;
	elseif (t_0 <= -0.35)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + ((J * (l * (K ^ 2.0))) * -0.25);
	tmp = 0.0;
	if (t_0 <= -0.96)
		tmp = t_1;
	elseif (t_0 <= -0.85)
		tmp = U;
	elseif (t_0 <= -0.35)
		tmp = t_1;
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.96], t$95$1, If[LessEqual[t$95$0, -0.85], U, If[LessEqual[t$95$0, -0.35], t$95$1, N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.95999999999999996 or -0.849999999999999978 < (cos.f64 (/.f64 K 2)) < -0.34999999999999998

   1. Initial program 80.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 53.4%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. *-commutative 53.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)} + U \]

      2. associate-*l* 53.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)} + U \]

      3. *-commutative 53.3%

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

   4. Simplified 53.3%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   5. Taylor expanded in K around 0 44.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)}\right) + U \]

   6. Taylor expanded in K around inf 57.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 57.7%

         \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25} + U \]

      2. *-commutative 57.7%

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot -0.25 + U \]

   8. Simplified 57.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25} + U \]

   if -0.95999999999999996 < (cos.f64 (/.f64 K 2)) < -0.849999999999999978

   1. Initial program 73.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in J around 0 54.9%

      \[\leadsto \color{blue}{U} \]

   if -0.34999999999999998 < (cos.f64 (/.f64 K 2))

   1. Initial program 88.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 60.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)} + U \]

      2. associate-*l* 60.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)} + U \]

      3. *-commutative 60.2%

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

   4. Simplified 60.2%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   5. Taylor expanded in K around 0 55.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 55.9%

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

      2. associate-*r* 56.4%

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   7. Simplified 56.4%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.96:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.85:\\ \;\;\;\;U\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.35:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.96:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{elif}\;t_0 \leq 0.21:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.96)
     (+ U (* (* J (* l (pow K 2.0))) -0.25))
     (if (<= t_0 0.21)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.96) {
		tmp = U + ((J * (l * pow(K, 2.0))) * -0.25);
	} else if (t_0 <= 0.21) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.96d0)) then
        tmp = u + ((j * (l * (k ** 2.0d0))) * (-0.25d0))
    else if (t_0 <= 0.21d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.96) {
		tmp = U + ((J * (l * Math.pow(K, 2.0))) * -0.25);
	} else if (t_0 <= 0.21) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.96:
		tmp = U + ((J * (l * math.pow(K, 2.0))) * -0.25)
	elif t_0 <= 0.21:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.96)
		tmp = Float64(U + Float64(Float64(J * Float64(l * (K ^ 2.0))) * -0.25));
	elseif (t_0 <= 0.21)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.96)
		tmp = U + ((J * (l * (K ^ 2.0))) * -0.25);
	elseif (t_0 <= 0.21)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.96], N[(U + N[(N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.21], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.95999999999999996

   1. Initial program 80.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 32.4%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. *-commutative 32.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)} + U \]

      2. associate-*l* 32.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)} + U \]

      3. *-commutative 32.6%

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

   4. Simplified 32.6%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   5. Taylor expanded in K around 0 61.3%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)}\right) + U \]

   6. Taylor expanded in K around inf 80.5%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 80.5%

         \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25} + U \]

      2. *-commutative 80.5%

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot -0.25 + U \]

   8. Simplified 80.5%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25} + U \]

   if -0.95999999999999996 < (cos.f64 (/.f64 K 2)) < 0.209999999999999992

   1. Initial program 83.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 68.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if 0.209999999999999992 < (cos.f64 (/.f64 K 2))

   1. Initial program 88.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 89.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in K around 0 85.9%

      \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.96:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.21:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 6: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot t_0\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.3:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.0135:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (* J 0.016666666666666666) (* (pow l 5.0) t_0))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -6.5e+65)
     t_1
     (if (<= l -2.3)
       t_2
       (if (<= l 0.0135)
         (+ U (* J (* (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)) t_0)))
         (if (<= l 2.8e+60) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + ((J * 0.016666666666666666) * (pow(l, 5.0) * t_0));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -6.5e+65) {
		tmp = t_1;
	} else if (l <= -2.3) {
		tmp = t_2;
	} else if (l <= 0.0135) {
		tmp = U + (J * (((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)) * t_0));
	} else if (l <= 2.8e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((j * 0.016666666666666666d0) * ((l ** 5.0d0) * t_0))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-6.5d+65)) then
        tmp = t_1
    else if (l <= (-2.3d0)) then
        tmp = t_2
    else if (l <= 0.0135d0) then
        tmp = u + (j * (((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)) * t_0))
    else if (l <= 2.8d+60) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + ((J * 0.016666666666666666) * (Math.pow(l, 5.0) * t_0));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -6.5e+65) {
		tmp = t_1;
	} else if (l <= -2.3) {
		tmp = t_2;
	} else if (l <= 0.0135) {
		tmp = U + (J * (((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)) * t_0));
	} else if (l <= 2.8e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + ((J * 0.016666666666666666) * (math.pow(l, 5.0) * t_0))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -6.5e+65:
		tmp = t_1
	elif l <= -2.3:
		tmp = t_2
	elif l <= 0.0135:
		tmp = U + (J * (((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)) * t_0))
	elif l <= 2.8e+60:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(Float64(J * 0.016666666666666666) * Float64((l ^ 5.0) * t_0)))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -6.5e+65)
		tmp = t_1;
	elseif (l <= -2.3)
		tmp = t_2;
	elseif (l <= 0.0135)
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)) * t_0)));
	elseif (l <= 2.8e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((J * 0.016666666666666666) * ((l ^ 5.0) * t_0));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -6.5e+65)
		tmp = t_1;
	elseif (l <= -2.3)
		tmp = t_2;
	elseif (l <= 0.0135)
		tmp = U + (J * (((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)) * t_0));
	elseif (l <= 2.8e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(J * 0.016666666666666666), $MachinePrecision] * N[(N[Power[l, 5.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -6.5e+65], t$95$1, If[LessEqual[l, -2.3], t$95$2, If[LessEqual[l, 0.0135], N[(U + N[(J * N[(N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e+60], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5000000000000003e65 or 2.8e60 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -6.5000000000000003e65 < l < -2.2999999999999998 or 0.0134999999999999998 < l < 2.8e60

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -2.2999999999999998 < l < 0.0134999999999999998

   1. Initial program 71.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 99.8%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+65}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -2.3:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.0135:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+60}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot t_0\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.00082:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.0072:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (* J 0.016666666666666666) (* (pow l 5.0) t_0))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -6.5e+65)
     t_1
     (if (<= l -0.00082)
       t_2
       (if (<= l 0.0072)
         (+ U (* 2.0 (* J (* l t_0))))
         (if (<= l 2.6e+60) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + ((J * 0.016666666666666666) * (pow(l, 5.0) * t_0));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -6.5e+65) {
		tmp = t_1;
	} else if (l <= -0.00082) {
		tmp = t_2;
	} else if (l <= 0.0072) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 2.6e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((j * 0.016666666666666666d0) * ((l ** 5.0d0) * t_0))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-6.5d+65)) then
        tmp = t_1
    else if (l <= (-0.00082d0)) then
        tmp = t_2
    else if (l <= 0.0072d0) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else if (l <= 2.6d+60) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + ((J * 0.016666666666666666) * (Math.pow(l, 5.0) * t_0));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -6.5e+65) {
		tmp = t_1;
	} else if (l <= -0.00082) {
		tmp = t_2;
	} else if (l <= 0.0072) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 2.6e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + ((J * 0.016666666666666666) * (math.pow(l, 5.0) * t_0))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -6.5e+65:
		tmp = t_1
	elif l <= -0.00082:
		tmp = t_2
	elif l <= 0.0072:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif l <= 2.6e+60:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(Float64(J * 0.016666666666666666) * Float64((l ^ 5.0) * t_0)))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -6.5e+65)
		tmp = t_1;
	elseif (l <= -0.00082)
		tmp = t_2;
	elseif (l <= 0.0072)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (l <= 2.6e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((J * 0.016666666666666666) * ((l ^ 5.0) * t_0));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -6.5e+65)
		tmp = t_1;
	elseif (l <= -0.00082)
		tmp = t_2;
	elseif (l <= 0.0072)
		tmp = U + (2.0 * (J * (l * t_0)));
	elseif (l <= 2.6e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(J * 0.016666666666666666), $MachinePrecision] * N[(N[Power[l, 5.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -6.5e+65], t$95$1, If[LessEqual[l, -0.00082], t$95$2, If[LessEqual[l, 0.0072], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+60], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5000000000000003e65 or 2.60000000000000008e60 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -6.5000000000000003e65 < l < -8.1999999999999998e-4 or 0.0071999999999999998 < l < 2.60000000000000008e60

   1. Initial program 99.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 99.6%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -8.1999999999999998e-4 < l < 0.0071999999999999998

   1. Initial program 70.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+65}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00082:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.0072:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+60}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00102 \lor \neg \left(\ell \leq 0.00098\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.00102) (not (<= l 0.00098)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00102) || !(l <= 0.00098)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.00102d0)) .or. (.not. (l <= 0.00098d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00102) || !(l <= 0.00098)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.00102) or not (l <= 0.00098):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.00102) || !(l <= 0.00098))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.00102) || ~((l <= 0.00098)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.00102], N[Not[LessEqual[l, 0.00098]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.00102 or 9.7999999999999997e-4 < l

   1. Initial program 99.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 80.1%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -0.00102 < l < 9.7999999999999997e-4

   1. Initial program 70.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00102 \lor \neg \left(\ell \leq 0.00098\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 9: 62.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+31} \lor \neg \left(\ell \leq 3.2 \cdot 10^{+181}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l 5e+31) (not (<= l 3.2e+181)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (pow U -4.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 5e+31) || !(l <= 3.2e+181)) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = pow(U, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= 5d+31) .or. (.not. (l <= 3.2d+181))) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 5e+31) || !(l <= 3.2e+181)) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= 5e+31) or not (l <= 3.2e+181):
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= 5e+31) || !(l <= 3.2e+181))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= 5e+31) || ~((l <= 3.2e+181)))
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, 5e+31], N[Not[LessEqual[l, 3.2e+181]], $MachinePrecision]], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000027e31 or 3.2e181 < l

   1. Initial program 84.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 67.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if 5.00000000000000027e31 < l < 3.2e181

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{{U}^{-4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+31} \lor \neg \left(\ell \leq 3.2 \cdot 10^{+181}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \]

Alternative 10: 62.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+31}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+180}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1e+31)
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
   (if (<= l 9.5e+180)
     (pow U -4.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1e+31) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 9.5e+180) {
		tmp = pow(U, -4.0);
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 1d+31) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if (l <= 9.5d+180) then
        tmp = u ** (-4.0d0)
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1e+31) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 9.5e+180) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1e+31:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif l <= 9.5e+180:
		tmp = math.pow(U, -4.0)
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1e+31)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 9.5e+180)
		tmp = U ^ -4.0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1e+31)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif (l <= 9.5e+180)
		tmp = U ^ -4.0;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1e+31], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.5e+180], N[Power[U, -4.0], $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.9999999999999996e30

   1. Initial program 83.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 70.6%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   3. Step-by-step derivation

      1. *-commutative 70.6%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot J\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*r* 71.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Simplified 71.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 9.9999999999999996e30 < l < 9.5000000000000003e180

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if 9.5000000000000003e180 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 38.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+31}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+180}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 11: 43.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+28} \lor \neg \left(\ell \leq 1220\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7.8e+28) (not (<= l 1220.0))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.8e+28) || !(l <= 1220.0)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-7.8d+28)) .or. (.not. (l <= 1220.0d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.8e+28) || !(l <= 1220.0)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7.8e+28) or not (l <= 1220.0):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7.8e+28) || !(l <= 1220.0))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -7.8e+28) || ~((l <= 1220.0)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7.8e+28], N[Not[LessEqual[l, 1220.0]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -7.7999999999999997e28 or 1220 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -7.7999999999999997e28 < l < 1220

   1. Initial program 72.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in J around 0 65.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+28} \lor \neg \left(\ell \leq 1220\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 12: 43.3% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -65000000000 \lor \neg \left(\ell \leq 720\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -65000000000.0) (not (<= l 720.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -65000000000.0) || !(l <= 720.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-65000000000.0d0)) .or. (.not. (l <= 720.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -65000000000.0) || !(l <= 720.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -65000000000.0) or not (l <= 720.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -65000000000.0) || !(l <= 720.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -65000000000.0) || ~((l <= 720.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -65000000000.0], N[Not[LessEqual[l, 720.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.5e10 or 720 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{U \cdot U} \]

   if -6.5e10 < l < 720

   1. Initial program 72.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in J around 0 67.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -65000000000 \lor \neg \left(\ell \leq 720\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 53.3% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot 2\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J (* l 2.0))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * 2.0));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * 2.0d0))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * 2.0));
}

Python

def code(J, l, K, U):
	return U + (J * (l * 2.0))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * 2.0)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * 2.0));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Taylor expanded in l around 0 60.5%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Step-by-step derivation

   1. *-commutative 60.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)} + U \]

   2. associate-*l* 60.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right)} + U \]

   3. *-commutative 60.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

4. Simplified 60.5%

   \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

5. Taylor expanded in K around 0 50.4%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} + U \]
6. Step-by-step derivation

   1. *-commutative 50.4%

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} + U \]

   2. associate-*r* 50.8%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

7. Simplified 50.8%

   \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

8. Final simplification 50.8%

   \[\leadsto U + J \cdot \left(\ell \cdot 2\right) \]

Alternative 14: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 1.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 1.0;
}

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

1. Initial program 86.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

3. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\frac{U}{U}} \]
4. Step-by-step derivation

   1. *-inverses 2.7%

      \[\leadsto \color{blue}{1} \]

5. Simplified 2.7%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 2.7%

   \[\leadsto 1 \]

Alternative 15: 37.2% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 86.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Taylor expanded in J around 0 32.6%

   \[\leadsto \color{blue}{U} \]

3. Final simplification 32.6%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023336 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))